REPORT ON CERTAIN BRANCHES OF ANALYSIS. 253 



iional. If we successively replace, therefore, x by -3- + i" and 

 •^ — X, we shall get 



It X 1 . _ 1 „ 



r * + -r + -77 = cos X + -TT- sin 2 J" 5- cos 3 x 



— 3- sin 4 X + &c. 

 r' IT + ^ ^ = cos 4: — -g- sin 2 x 5- cos 3 a? 



+ -p sin 4 :r + &c. 



Adding these two series together and dividing by 2, we get 



{r 4- i^\ It 1 1 



^ g ' ir + -J- = cos .r — -^ cos 3ar + -^ cos 5 ;r — &c. (2.) 



It It 

 If a; be included between -^ and g-, then r = and r' = 0, 



and we get 



It I I 



-J- = cos X 5- COS 3 ^ + -^ cos 5 ^ — &c. (3.) 



If xbe included between -5- and -3-, thenr = — 1 and r* = 0, 



and we get 



« 1 1 



— -^ = COS X jT COS 3 .r + "TT cos 5x — &c. (4.) 



4 5 ^ ' 



If the limits of x be -^ and -^, — ^ and -^, -^ and 

 — -^, ^ and ^, we shall obtain values of the series 



(2.), which are alternately -j- and — -j- 



2 ~ ■'"' *" 2 



X . I . 



Again, if in equation (1.), or rtt -i- — = sin a? — ^-sin^jr 



+ -5- sin 3 a: j- sin 4 a* + &c., we replace xhy tt — x, we 



shall get 



11 X . 1 1 . 1 . 



r' Tt -\ — = sin^ + -Q-sin 2x + -3- sin 3 J? + -x-sin4x + &c. 



Adding these equations together and dividing by 2, we get 



